Electronic transport in GaAs

Chapter III: First-principles simulations of nonequilibrium electronic fluctu-

3.1 Electronic transport in GaAs

Gallium arsenide (GaAs) is III-V compound semiconductor with zincblende crystal structure. As depicted in Fig.3.1, GaAs is a direct bandgap semiconductor with a single conduction band minimum at theΓ point. To excellent approximation, the Gamma valley is spherical with effective mass 0.063𝑚

0 [86]. Because of the high mobility resulting from a light conduction band minimum, GaAs is a commonly used material in high speed electronics operating at microwave frequencies [38].

Further up the conduction band, GaAs possesses eight degenerate valleys at the 𝐿 point and six degenerate valleys at the𝑋point. The large energy separation between the valleys Δ𝐸_Γ𝐿 = 300 meV and Δ𝐸_Γ𝑋 = 470 meV [145] means that at room temperature and for typical doping levels, equilibrium electrons remain mostly in theΓvalley (for 300 K and𝑛=10¹⁶ cm⁻³, over 99.9% of electrons are inΓ).

We begin this section examining the steady state distribution and associated transport observables for the cold and warm electron regimes in room temperature GaAs.

Figure3.2 plots the deviational steady state distribution functions Δ𝑓_k = 𝑓^𝑠

k − 𝑓⁰

k

under the two approximations versus the wave vector parallel to the electric field,𝑘_𝑥. We refer to this direction as the longitudinal direction. At low fieldsE <100 V cm⁻¹, the cold electron approximation is accurate and the solutions are nearly identical.

Figure 3.1: Conduction and valence band structure of GaAs with the 𝐹43¯ 𝑚[216] Hermann Mauguin space group. Inset: Conventional cell of GaAs in the zincblende structure. Data generated with the Materials Project [144].

In contrast, as the field increases, differences in the distribution functions emerge.

Under the cold electron approximation, Eqn. 2.12 shows that Δ𝑓_k is required to possess odd symmetry about the Brillouin zone center because the forcing gradient

𝜕 𝑓⁰

k/𝜕_kis odd with respect to𝑘_𝑥while the scattering matrix is even (Θ_kk⁰ = Θ−k−k⁰).

This symmetry is evident in the cold electron solutions at all fields in Fig. 3.2.

Physically, this symmetry indicates that in the cold electron case, states with negative wavevector are depopulated to the exact same degree that the corresponding positive wavevector are filled. Since the net population does not change with energy, electrons under this approximation do not heat with the field. This is in contrast to the warm electron case, where the electronic distribution function develop asymmetrically in momentum space as the non-equilibrium gradient modeled by Eqn. 2.8 grows with the field. At E = 800 V cm⁻¹, warm electrons reach comparatively high wavevector states in the direction of the field as exhibited by the momentum-space

Figure 3.2: Deviational occupation Δ𝑓_𝑘 in GaAs at 300 K under the cold (dotted lines) and warm (solid lines) electron approximations versus longitudinal wave vector 𝑘_𝑥. Curves plotted for E = 100 V cm⁻¹ (blue), and E = 800 V cm⁻¹ (orange). The dashed black line is a guide to the eye. At the lower field, the two approximations result in nearly identical distribution functions. At the higher field, the cold-electron approximation fails to capture the assymetrical development of the distribution function.

tail in Fig.3.2. These electrons absorb energy from the field and are heated above the lattice temperature.

The transport properties of the warm electron distribution differ from those of the cold distribution because warm electrons in the high momentum, high energy tail are able to emit optical phonons and hence exhibit higher scattering rates. As reported previously [92], the predicted mobility of GaAs exceeds the experimental mobility owing to the exclusion of higher-order phonon scattering processes and the lower calculated effective mass (0.055𝑚

0) compared to experiment (0.067𝑚

0) [93].

Therefore, to facilitate comparison, we examine the DC mobility normalized by its low-field value in Fig.3.3. The low-field value of the computed mobility is 17,420

Figure 3.3: Normalized longitudinal (k) DC mobility versus electric field of the cold (dashed blue line) and warm electrons (solid red line). The heating of the electrons leads to a decreased mobility. The trend of the normalized mobility agrees well with experiments: Figure 1, Ref. [146] (Upward black triangles) and Figure 4, Ref. [147]

(Downward black triangles).

cm²V⁻¹s⁻¹. At low fieldsE < 100 V cm⁻¹, the mobility under the warm and cold electron approximations agrees to within 1%. At higher fieldsE =800 V cm⁻¹, the DC mobility of the warm electrons has decreased by more than 10%. This behavior is qualitatively consistent with the sublinear current voltage characteristic (CVC) of n-type GaAs [38], or a decrease in mobility with increasing electron temperature caused by a concomitant increase in the average scattering rate. The field dependence of the normalized mobility shows favorable comparison to experiment, implying that our calculation is properly capturing the heating with the field.

In addition to steady quantities, the small-signal AC mobility can be computed as in Eqn.2.17. Figure3.4 presents the small-signal AC mobility for the warm electron gas versus frequency for several electric fields. At zero frequency, the equilibrium AC mobility is equal to the equilibrium DC mobility, as expected. The decrease of

Figure 3.4: Real part of the longitudinal small-signal AC mobility versus frequency for equilibrium (dashed black line), E = 100 V cm⁻¹ (dash-dot blue line), and E = 800 V cm⁻¹ (solid orange line) under the warm electron approximation. The AC mobility exhibits spectral features at frequencies that are characteristic of the inverse momentum and energy relaxation times (see Section3.3).

the AC mobility with electric field is also consistent with the trend observed in the DC mobility. At 𝑓 ∼1 THz, the AC field frequency exceeds the phonon-mediated scattering rates which redistribute the electrons in response to the field, and thus the AC mobility rolls off at all fields. This result reflects the electrical response transitioning from a purely resistive to a purely reactive regime as the frequency exceeds the highest scattering rates.

The frequency dependence of the AC mobility indicates the relevant timescales of momentum and energy relaxation [148]. In particular, for 800 V cm⁻¹, we observe a lower value of the AC mobility at low frequency, followed by a maximum at around 100 GHz. This feature is due to energy exchange with phonons and will be discussed in Section3.3.

Figure 3.5: Spectral density of longitudinal current density fluctuations (solid red line) normalized to the Nyquist value versus electric field along with Davydov spectral densities calculated using an acoustic deformation potential (ADP) (dash- dot blue line) and Fröhlich (dashed yellow line). At equilibrium, the noise agrees with Nyquist-Johnson noise (dotted black line). The ab initiocalculation predicts a steeper decrease in current PSD with field compared to the approximations. The symbols correspond to experimental measurements (Figure 11, Ref. [149]).